NationMaster - Encyclopedia: Hooke's law

In physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress). Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials. Fluid mechanics is the subdiscipline of continuum mechanics that studies fluids, that is, liquids and gases. ... A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... Fluid statics (also called hydrostatics) is the science of fluids at rest, and is a sub-field within fluid mechanics. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... For other uses, see Viscosity (disambiguation). ... A Newtonian fluid (named for Isaac Newton) is a fluid that flows like waterâ€”its shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. ... A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. ... Surface tension is an effect within the surface layer of a liquid that causes that layer to behave as an elastic sheet. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Sir George Gabriel Stokes, 1st Baronet (13 August 1819â€“1 February 1903) was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics (including the Navier-Stokes equations), optics, and mathematical physics (including Stokes theorem). ... Claude-Louis Navier (born Claude Louis Marie Henri Navier on February 10, 1785 in Dijon, died August 21, 1836 in Paris) was a French engineer and physicist. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Solid mechanics (also known as the theory of elasticity) is a branch of physics, which governs the response of solid material to applied stress (e. ... This article is about the deformation of materials. ... Stress is a measure of force per unit area within a body. ... // Linear elasticity The linear theory of elasticity models the macroscopic mechanical properties of solids assuming small deformations. ...

Hooke's law is named after the 17^th century British physicist Robert Hooke. He first stated this law in 1676 as an anagram[1], then in 1678 in Latin as Ut tensio, sic vis, which means: (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... Robert Hooke, FRS (July 18, 1635 â€“ March 3, 1703) was an English polymath who played an important role in the scientific revolution, through both experimental and theoretical work. ...

For systems that obey Hooke's law, the extension produced is directly proportional to the load:

When this holds, we say that the behavior is linear. If shown on a graph, the line should show a direct variation. There is a negative sign on the right hand side of the equation because the restoring force always acts in the opposite direction of the x displacement (when a spring is stretched to the left, it pulls back to the right). This article is about the SI unit of force. ... The metre, or meter (symbol: m) is the SI base unit of length. ... This article is about proportionality, the mathematical relation. ...

Elastic materials

Objects that quickly regain their original shape after being deformed by a stress, with the molecules or atoms of their material returning to the initial state of stable equilibrium, often obey Hooke's law.

We may view a rod of any elastic material as a linear spring. The rod has length L and cross-sectional area A. Its extension (strain) is linearly proportional to its tensile stress, σ by a constant factor, the inverse of its modulus of elasticity, E, hence, Elasticity is a branch of physics which studies the properties of elastic materials. ... Tensile stress (or tension) is the stress state leading to expansion; that is, the length of a material tends to increase in the tensile direction. ... In solid mechanics, Youngs modulus (also known as the modulus of elasticity or elastic modulus) is a measure of the Stiffness of a given material. ...

Hooke's law only holds for some materials under certain loading conditions. Steel exhibits linear-elastic behavior in most engineering applications; Hooke's law is valid for it throughout its elastic range (i.e., for stresses below the yield strength). For some other materials, such as aluminium, Hooke's law is only valid for a portion of the elastic range. For these materials a proportional limit stress is defined, below which the errors associated with the linear approximation are negligible. For other uses, see Steel (disambiguation). ... Yield strength, or the yield point, is defined in engineering and materials science as the stress at which a material begins to plastically deform. ... Aluminum redirects here. ... The proportional limit is the maximum stress a material can undergo where the relationship between stress and strain are linearly proportional. ...

Rubber is generally regarded as a "non-hookean" material because its elasticity is stress dependent and sensitive to temperature and loading rate. This does not cite any references or sources. ...

Applications of the law include spring operated weighing machines, stress analysis and modeling of materials.

The spring equation

The most commonly encountered form of Hooke's law is probably the spring equation, which relates the force exerted by a spring to the distance it is stretched by a spring constant, k, measured in force per length. Image File history File links Stress_v_strain_A36_2. ... Image File history File links Stress_v_strain_A36_2. ... A stress-strain curve is a graph derived from measuring load (stress - Ïƒ) versus extension (strain - Îµ) for a sample of a material. ... For other uses, see Steel (disambiguation). ... Yield strength, or the yield point, is defined in engineering as the amount of strain that a material can undergo before moving from elastic deformation into plastic deformation. ... Cold Work is a quality imparted on a material as a result of plastic deformation. ...

The negative sign indicates that the force exerted by the spring is in direct opposition to the direction of displacement. It is called a "restoring force", as it tends to restore the system to equilibrium.

which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over distance. (Note that potential energy of a spring is always positive.)

This potential can be visualized as a parabola on the U-x plane. As the spring is stretched in the positive x-direction, the potential energy increases (the same thing happens as the spring is compressed). The corresponding point on the potential energy curve is higher than that corresponding to the equilibrium position (x = 0). The tendency for the spring is to therefore decrease its potential energy by returning to its equilibrium (unstretched) position, just as a ball rolls downhill to decrease its gravitational potential energy.

If a mass m is attached to the end of such a spring, the system becomes a harmonic oscillator. It will oscillate with a natural frequency given as either: Vibration and standing waves in a string, The fundamental and the first 6 overtones The fundamental tone, often referred to simply as the fundamental and abbreviated fo, is the lowest frequency in a harmonic series. ...

where

ν

is frequency (the symbol is the Greek character nu and not the letter v) since $,omega = {2 pi nu}$ .

Multiple springs

When two springs are attached to a mass and compressed, the following table compares values of the springs.

Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

Derivation

Equivalent Spring Constant (Series)

Deriving $k e q$ in the series case is a little trickier than in the parallel case. Defining the equilibrium position of the block to be $x 2$ , we'll be looking for equation for the force on the block that looks like:

$F_b = -k_{eq} x_2 .,$

To begin, we'll also define the equilibrium position of the point between the two springs to be $x 1$ .

The force on the block is

$F_b = - k_2 left( x_2 - x_1 right). quad quad quad (1) ,$

Meanwhile, the force on the point between the two springs is

$F_s = - k_1 x_1 + k_2 (x_2 - x_1). ,$

Now, when the block is pushed so the springs are compressed and the system is allowed to come to equilibrium, the force between the springs must sum to zero, so with $F s = 0$ we can solve for $x_1 ,$ :

$- k_1 x_1 + k_2 (x_2 - x_1) = 0 ,$

$- k_1 x_1 - k_2 x_1 = -k_2 x_2 ,$

$left(k_1 + k_2 right) x_1 = k_2 x_2 ,$

$x_1 = frac{k_2}{k_1 + k_2} x_2 . ,$

Now we just plug this back into (1):

$F_b ,$	$= -k_2 x_2 + k_2 x_1 ,$
	$= -k_2 x_2 + k_2 left( frac{k_2}{k_1 + k_2} x_2 right) ,$
	$= -k_2 x_2 left( frac{k_1 + k_2}{k_1 + k_2} right) + frac{k_2^2}{k_1 + k_2} x_2 ,$
	$= x_2 frac{-k_1 k_2 - k_2^2 + k_2^2}{k_1 + k_2} ,$

Finally, the force on the block has been found:

$F_b = - left( frac{k_1 k_2 }{k_1 + k_2} right) x_2 .,$

So we can define everything in the parenthesis to be

$k_{eq} = frac{k_1 k_2 }{k_1 + k_2} .,$

Which can also be written:

$frac{1}{k_{eq}} = frac{1}{k_1} + frac{1}{k_2}. ,$

Equivalent Spring Constant (Parallel)

Both springs are touching the block in this case, and whatever distance spring 1 is compressed has to be the same amount spring 2 is compressed.

The force on the block is then:

$F_b ,$	$= F_1 + F_2 ,$
	$= -k_1 x - k_2 x ,$

So the force on the block is

$F_b = - (k_1 + k_2) x. ,$

Which is why we can define the equivalent spring constant as

$k_{eq} = k_1 + k_2 . ,$

Compressed Distance

In the case where two springs are in series, the magnitude of the force of the springs on each other are equal:

$|F_1| = |F_2| ,$

$k_1 x_1 = k_2 left(x_2 -x_1 right). ,$

For spring 1, x₁ is the distance from equilibrium length, and for spring 2, x₂ - x₁ is the distance from its equilibrium length. So we can define

$a_1 = x_1 ,$

$a_2 = x_2 - x_1. ,$

Plug these definitions into the force equation, and we'll get a relationship between the compresed distances for the in series case:

$frac{a_1}{a_2} = frac{k_2}{k_1}. ,$

Energy Stored

For the series case, the ratio of energy stored in springs is:

$frac{E_1}{E_2} = frac{frac{1}{2} k_1 a_1^2}{frac{1}{2}k_2 a_2^2}, ,$

but a there is a relationship between a₁ and a₂ derived earlier, so we can plug that in:

$frac{E_1}{E_2} = frac{k_1}{k_2} left(frac{k_2}{k_1}right)^2 = frac{k_2}{k_1} . ,$

For the parallel case,

$frac{E_1}{E_2} = frac{frac{1}{2} k_1 x^2}{frac{1}{2}k_2 x^2} ,$

because the compressed distance of the springs is the same, this simplifies to

$frac{E_1}{E_2} = frac{k_1}{k_2}. ,$

Tensor expression of Hooke's Law

When working with a three-dimensional stress state, a 4^th order tensor (c_ijkl) containing 81 elastic coefficients must be defined to link the stress tensor (σ_ij) and the strain tensor (or Green tensor) (ε_kl). In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... This article is in need of attention from an expert on the subject. ... The strain tensor, Îµ, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: the diagonal coefficients Îµii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms Îµij = 1/2 Î³ij (i... The strain tensor [ε] is a symmetric tensor used to quantify the strain of an object undergoing a 3-dimensional deformation: the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms εij (i &#8800...

Due to the symmetry of the stress tensor, strain tensor, and stiffness tensor, only 21 elastic coefficients are independent. Stiffness is the resistance of an elastic body to deflection or deformation by an applied force. ...

As stress is measured in units of pressure and strain is dimensionless, the entries of c_ijkl are also in units of pressure.

Generalization for the case of large deformations is provided by models of neo-Hookean solids and Mooney-Rivlin solids. In engineering mechanics, deformation is a change in shape due to an applied force. ... Neo-Hookean solid model is an extension of Hookes law for the case of large deformations. ... In continuum mechanics, a Mooney-Rivlin solid is a generalization of the Neo-Hookean solid model, where the strain energy W is a linear combination of two invariants of Finger tensor : , where and are the first and the second invariant of Finger tensor, and , are constants. ...

Isotropic materials

(see viscosity for an analogous development for viscous fluids.) For other uses, see Viscosity (disambiguation). ...

Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials must therefore be independent of the coordinate system chosen to represent them. The strain tensor is a symmetric tensor. Since the trace of any tensor is independent of coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor. (Symon (1971) Ch. 10) Thus: Look up Trace in Wiktionary, the free dictionary. ...

where

δ i j

is the Kronecker delta. The first term on the right is the constant tensor, also known as the pressure, and the second term is the traceless symmetric tensor, also known as the shear tensor. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... This article is about pressure in the physical sciences. ...

The most general form of Hooke's law for isotropic materials may now be written as a linear combination of these two tensors:

where K is the bulk modulus and G is the shear modulus. The bulk modulus (K) of a substance essentially measures the substances resistance to uniform compression. ... In materials science, shear modulus, G, or sometimes S or Î¼, sometimes referred to as the modulus of rigidity, is defined as the ratio of shear stress to the shear strain:[1] where = shear stress; force acts on area ; = shear strain; length changes by amount . ...

Using the relationships between the elastic moduli, these equations may also be expressed in various other ways. For example, the strain may be expressed in terms of the stress tensor as: An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substances tendency to be deformed when a force is applied to it. ...

where Y is the modulus of elasticity and

ν

is Poisson's ratio. (See 3-D elasticity). In solid mechanics, Youngs modulus (E) is a measure of the stiffness of a given material. ... 3-D elasticity is one of three methods of structural analysis. ...

Derivation of Hooke's law in 3D

The 3-D form of Hooke's law can be derived using Poisson's ratio and the 1-D form of Hooke's law as follows. Consider the strain and stress relation as a superposition of two effects: stretching in direction of the load (1) and shrinking (caused by the load) in perpendicular directions (2 and 3),

$varepsilon_1' = frac{1}{E}sigma_1$ ,

$varepsilon_2' = -nuvarepsilon_1$ ,

$varepsilon_3' = -nuvarepsilon_1$ ,

where $ν$ is the Poisson's ratio and $E$ the Young Modulus. We get similar equations to the loads in directions 2 and 3, Figure 1: Rectangular specimen subject to compression, with Poissons ratio circa 0. ... This article does not adequately cite its references or sources. ...

$varepsilon_1'' = -nuvarepsilon_2$ ,

$varepsilon_2'' = frac{1}{E}sigma_2$ ,

$varepsilon_3'' = -nuvarepsilon_2$ ,

and

$varepsilon_1''' = -nuvarepsilon_3$ ,

$varepsilon_2''' = -nuvarepsilon_3$ ,

$varepsilon_3''' = frac{1}{E}sigma_3$ .

Summing the three cases together ( $varepsilon_i = varepsilon_i' + varepsilon_i'' +varepsilon_i'''$ ) we get

$varepsilon_1 = frac{1}{E}(sigma_1-nu(sigma_2+sigma_3))$

$varepsilon_2 = frac{1}{E}(sigma_2-nu(sigma_1+sigma_3))$

$varepsilon_3 = frac{1}{E}(sigma_3-nu(sigma_1+sigma_2))$

or by adding and subtracting one $νσ$

$varepsilon_1 = frac{1}{E}((1+nu)sigma_1-nu(sigma_1+sigma_2+sigma_3))$

$varepsilon_2 = frac{1}{E}((1+nu)sigma_2-nu(sigma_1+sigma_2+sigma_3))$

$varepsilon_3 = frac{1}{E}((1+nu)sigma_3-nu(sigma_1+sigma_2+sigma_3))$

and further we get by solving $σ 1$

$sigma_1 = frac{E}{1+nu}varepsilon_1 + frac{nu}{1+nu}(sigma_1+sigma_2+sigma_3)$ .

Calculating the sum

$sum_{i=1,2,3}varepsilon_i = frac{1}{E}((1+nu)sum_{i=1,2,3}sigma_i - 3nu(sum_{i=1,2,3}sigma_i)) = frac{1-2nu}{E}sum_{i=1,2,3}sigma_i$

$sigma_1 +sigma_2+sigma_3 = frac{E}{1-2nu}(varepsilon_1 + varepsilon_2 +varepsilon_3)$

and substituting it to the equation solved for $σ 1$ gives

$sigma_1 = frac{E}{1+nu}varepsilon_1 + frac{Enu}{(1+nu)(1-2nu)}(varepsilon_1 + varepsilon_2 +varepsilon_3)$ ,

$sigma_1 = 2muvarepsilon_1 + lambda(varepsilon_1 + varepsilon_2 +varepsilon_3)$ ,

where $μ$ and $λ$ are the Lamé parameters. Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions. In linear elasticity, the LamÃ© parameters are the two parameters which in homogenous, isotropic materials satisfy the equation where is the stress and the strain tensor. ...

Zero-length springs

"Zero-length spring" is the standard term for a spring that exerts zero force when it has zero length. In practice this is done by combining a spring with "negative" length (in which the coils press together when the spring is relaxed) with an extra length of inelastic material. This type of spring was developed in 1932 by Lucien LaCoste for use in a vertical seismograph. A spring with zero length can be attached to a mass on a hinged boom in such a way that the force on the mass is almost exactly balanced by the vertical component of the force from the spring, whatever the position of the boom. This creates a pendulum with very long period. Long-period pendulums enable seismometers to sense the slowest waves from earthquakes. The LaCoste suspension with zero-length springs is also used in gravimeters because it is very sensitive to changes in gravity. Springs for closing doors are often made to have roughly zero length so that they will exert force even when the door is almost closed, so it will close firmly. LaCoste, Lucien, 1908-1995, scientist, inventor Many other articles will tell you the important things about Dr. LaCoste, but the purpose of this entry is to pass on two memories of a great man. ... Seismographs (in Greek seismos = earthquake and graphein = write) are used by seismologists to record seismic waves. ... Seismometers (in Greek seismos = earthquake and metero = measure) are used by seismologists to measure and record the size and force of seismic waves. ... An accelerometer or gravimeter is a device for measuring acceleration and the effects of gravity. ...